There are **12** NRICH Mathematical resources connected to **Proof by contradiction**, you may find related items under Thinking Mathematically.

Challenge Level

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

An introduction to proof by contradiction, a powerful method of mathematical proof.

This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.

Challenge Level

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

Challenge Level

Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.

Challenge Level

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

Challenge Level

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

Challenge Level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Challenge Level

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.